Standard mixed finite element methods for the incompressible Navier–Stokes equations that
relax the divergence constraint are not robust against large irrotational forces in the
momentum balance and the velocity error depends on the continuous pressure. This
robustness issue can be completely cured by using divergence-free mixed finite elements
which deliver pressure-independent velocity error estimates. However, the construction of
H1-conforming,
divergence-free mixed finite element methods is rather difficult. Instead, we present a
novel approach for the construction of arbitrary order mixed finite element methods which
deliver pressure-independent velocity errors. The approach does not change the trial
functions but replaces discretely divergence-free test functions in some operators of the
weak formulation by divergence-free ones. This modification is applied to inf-sup stable
conforming and nonconforming mixed finite element methods of arbitrary order in two and
three dimensions. Optimal estimates for the incompressible Stokes equations are proved for
the H1 and
L2 errors of the
velocity and the L2
error of the pressure. Moreover, both velocity errors are pressure-independent,
demonstrating the improved robustness. Several numerical examples illustrate the
results.

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